Interpolation theorems, lower Bounds for proof systems, and independence results for bounded arithmetic
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Journal of Symbolic Logic 62 (2):457-486 (1997)
A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1) Feasible interpolation theorems for the following proof systems: (a) resolution (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (α) (c) linear equational calculus (d) cutting planes. (2) New proofs of the exponential lower bounds (for new formulas) (a) for resolution () (b) for the cutting planes proof system with coefficients written in unary (). (3) An alternative proof of the independence result of  concerning the provability of circuit-size lower bounds in the bounded arithmetic theory S 2 2 (α). In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle
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Samuel R. Buss (2012). Towards–Via Proof Complexity and Search. Annals of Pure and Applied Logic 163 (7):906-917.
Pavel Hrubeš (2009). On Lengths of Proofs in Non-Classical Logics. Annals of Pure and Applied Logic 157 (2):194-205.
Pavel Hrubeš (2007). A Lower Bound for Intuitionistic Logic. Annals of Pure and Applied Logic 146 (1):72-90.
Arnold Beckmann & Samuel R. Buss (2005). Separation Results for the Size of Constant-Depth Propositional Proofs. Annals of Pure and Applied Logic 136 (1-2):30-55.
Ján Pich (2011). Nisan-Wigderson Generators in Proof Systems with Forms of Interpolation. Mathematical Logic Quarterly 57 (4):379-383.
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